Nuprl Lemma : factorit_wf
∀[x:ℕ+]. ∀[b:ℕ].
∀[tried:{L:{p:ℕ| prime(p) ∧ p < b} List| ∀p:{p:ℕ| prime(p)} . (p < b ⇒ ((p ∈ L) ∧ (¬(p | x))))} ].
∀[facs:{p:ℕ| prime(p)} List].
(factorit(x;b;tried;facs) ∈ {L:{p:ℕ| prime(p)} List| reduce(λp,q. (p * q);1;L) = (x * reduce(λp,q. (p * q);1;facs))\000C ∈ ℤ} )
supposing 2 ≤ b
Proof
Definitions occuring in Statement :
factorit: factorit(x;b;tried;facs),
prime: prime(a),
divides: b | a,
l_member: (x ∈ l),
reduce: reduce(f;k;as),
list: T List,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
factorit: factorit(x;b;tried;facs),
nat_plus: ℕ+,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
lt_int: i <z j,
reduce: reduce(f;k;as),
list_ind: list_ind,
cons: [a / b],
nequal: a ≠ b ∈ T ,
prime: prime(a),
int_nzero: ℤ-o,
iff: P ⇐⇒ Q,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
has-value: (a)↓,
l_exists: (∃x∈L. P[x]),
less_than: a < b,
squash: ↓T,
sq_stable: SqStable(P),
cand: A c∧ B,
l_member: (x ∈ l),
list: T List,
divides: b | a,
true: True,
subtract: n - m,
less_than': less_than'(a;b),
select: L[n]
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
list_wf,
nat_wf,
prime_wf,
less_than_wf,
l_member_wf,
subtype_rel_list,
divides_wf,
nat_plus_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
lt_int_wf,
equal-wf-base,
bool_wf,
le_wf,
assert_wf,
le_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
nat_plus_properties,
reduce_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
list_subtype_base,
cons_wf,
nat_plus_subtype_nat,
primality-test,
mul_preserves_le,
bl-exists_wf,
eq_int_wf,
remainder_wfa,
nequal_wf,
assert-bl-exists,
l_exists_functionality,
and_wf,
iff_weakening_uiff,
assert_of_eq_int,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
l_exists_wf,
value-type-has-value,
int-value-type,
prime_elim,
select_wf,
divides_iff_rem_zero,
int_nzero_wf,
length_wf,
sq_stable__all,
assoced_wf,
false_wf,
sq_stable_from_decidable,
decidable__false,
assoced_nelim,
l_exists_iff,
div_rem_sum,
not_wf,
istype-assert,
iff_transitivity,
assert_of_bnot,
divide_wfa,
div_bounds_1,
mul_preserves_lt,
add-is-int-iff,
multiply-is-int-iff,
divides_transitivity,
reduce_cons_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal,
subtype_rel_list_set,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
less-iff-le,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
cons_member,
all_wf,
set_wf,
satisfiable-full-omega-tt
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
productElimination,
because_Cache,
unionElimination,
applyEquality,
instantiate,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
setEquality,
setIsType,
productEquality,
functionIsType,
intEquality,
multiplyEquality,
addEquality,
baseApply,
closedConclusion,
baseClosed,
equalityElimination,
equalityIstype,
cumulativity,
sqequalBase,
promote_hyp,
callbyvalueReduce,
hyp_replacement,
imageElimination,
unionEquality,
unionIsType,
imageMemberEquality,
pointwiseFunctionality,
universeEquality,
minusEquality,
inrFormation_alt,
functionEquality,
lambdaFormation,
lambdaEquality,
isect_memberEquality,
isect_memberFormation,
computeAll,
voidEquality,
dependent_pairFormation,
dependent_set_memberEquality
Latex:
\mforall{}[x:\mBbbN{}\msupplus{}]. \mforall{}[b:\mBbbN{}].
\mforall{}[tried:\{L:\{p:\mBbbN{}| prime(p) \mwedge{} p < b\} List| \mforall{}p:\{p:\mBbbN{}| prime(p)\} . (p < b {}\mRightarrow{} ((p \mmember{} L) \mwedge{} (\mneg{}(p | x))))\} \000C].
\mforall{}[facs:\{p:\mBbbN{}| prime(p)\} List].
(factorit(x;b;tried;facs) \mmember{} \{L:\{p:\mBbbN{}| prime(p)\} List|
reduce(\mlambda{}p,q. (p * q);1;L) = (x * reduce(\mlambda{}p,q. (p * q);1;facs))\} )
supposing 2 \mleq{} b
Date html generated:
2019_10_15-AM-11_10_14
Last ObjectModification:
2019_06_25-PM-01_22_53
Theory : general
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